Answer :
Let's work through the solution step-by-step to find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex].
1. Given Information:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
2. Function Formula:
The function is given by:
[tex]\[
f(t) = P e^{rt}
\][/tex]
3. Set Up the Equation:
Substitute the known values into the function to find [tex]\( P \)[/tex]:
[tex]\[
f(4) = P e^{0.04 \times 4}
\][/tex]
[tex]\[
246.4 = P e^{0.16}
\][/tex]
4. Calculate [tex]\( e^{0.16} \)[/tex]:
Use the exponential function [tex]\( e^{0.16} \)[/tex] to find its value:
[tex]\[
e^{0.16} \approx 1.17351
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351}
\][/tex]
[tex]\[
P \approx 209.97
\][/tex]
6. Approximate [tex]\( P \)[/tex]:
Round [tex]\( P \)[/tex] to the nearest whole number and see which option it closely matches:
[tex]\[
P \approx 210
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 210, which corresponds to option C.
1. Given Information:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
2. Function Formula:
The function is given by:
[tex]\[
f(t) = P e^{rt}
\][/tex]
3. Set Up the Equation:
Substitute the known values into the function to find [tex]\( P \)[/tex]:
[tex]\[
f(4) = P e^{0.04 \times 4}
\][/tex]
[tex]\[
246.4 = P e^{0.16}
\][/tex]
4. Calculate [tex]\( e^{0.16} \)[/tex]:
Use the exponential function [tex]\( e^{0.16} \)[/tex] to find its value:
[tex]\[
e^{0.16} \approx 1.17351
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351}
\][/tex]
[tex]\[
P \approx 209.97
\][/tex]
6. Approximate [tex]\( P \)[/tex]:
Round [tex]\( P \)[/tex] to the nearest whole number and see which option it closely matches:
[tex]\[
P \approx 210
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 210, which corresponds to option C.
